Using Concepts Maps in a Foundation Mathematics Course: What Have we Learnt?

Sali Hammad; Christos Dimitriadis; Ted Graham

doi:10.29333/ejmste/9700

Full Text (PDF)

Sali Hammad ¹ ^* , Christos Dimitriadis ², Ted Graham ³

More Detail

¹ Gulf University for Science and Technology, KUWAIT² Kingston University, UK³ Independent Researcher, UK^* Corresponding Author

Abstract

This paper reports on the findings from the use of concept maps in a mathematics foundation course at a university in Kuwait. The study sample consisted of 130 freshmen students from a mathematics foundation course. Through a case-study design, concept maps and assessment tests were utilized to assess and monitor students’ mathematical understanding and achievement at various points of the course. The findings of the study showed that concept maps can be effective as a tool to assess, monitor, and improve students’ mathematical understanding, particularly their conceptual understanding when they are used systematically and when their use is followed-up by discussions that encourage students to reflect and talk about their maps and the links they have made. The improved understanding was found to contribute to the enhancement of mathematical achievement. The paper reports details on the use of concept maps in mathematics lessons and makes recommendations for practice and future research.

Keywords

mathematical understanding
concept maps
procedural understanding
conceptual understanding

License

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

EURASIA J Math Sci Tech Ed, Volume 17, Issue 2, February 2021, Article No: em1943

https://doi.org/10.29333/ejmste/9700

Publication date: 13 Feb 2021

Article Views: 2784

Article Downloads: 3188

Open Access Disclosures References How to cite this article

Disclosure

Declaration of Conflict of Interest: No conflict of interest is declared by author(s).

Data sharing statement: Data supporting the findings and conclusions are available upon request from the corresponding author(s).

References

Akkaya, R., Karakirik, E., & Durmus, S. (2005). A Computer Assessment Tool for Concept Mapping. Turkish Online Journal of Educational Technology-TOJET, 4(3), 3-6.
Apino, E., & Retnawati, H. (2017). Developing instructional design to improve mathematical higher order thinking skills of students. In Journal of Physics: Conference Series (Vol. 812, No. 1, p. 012100). IOP Publishing. https://doi.org/10.1088/1742-6596/812/1/012100
Baroody, A. J. (2003). The development of adaptive expertise and flexibility: The integration of conceptual and procedural knowledge. In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills: Constructing adaptive expertise (pp. 1-33). Mahwah, NJ: Erlbaum.
Berger, R. (2015). Now I see it, now I don’t: Researcher’s position and reflexivity in qualitative research. Qualitative Research, 15(2), 219-234. https://doi.org/10.1177/1468794112468475
Cliburn, J. W. (1990). Concept maps to promote meaningful learning. Journal of College Science Teaching, 19, 212-217.
Cohen, L., Manion, L., & Morrison, K. (2013). Research Methods in Education. London: Routledge. https://doi.org/10.4324/9780203720967
Davies, M. (2011). Concept mapping, mind mapping and argument mapping: what are the differences and do they matter?. Higher education, 62(3), 279-301. https://doi.org/10.1007/s10734-010-9387-6
DeCaro, M. S. (2016). Inducing mental set constrains procedural flexibility and conceptual understanding in mathematics. Memory & cognition, 44(7), 1138-1148. https://doi.org/10.3758/s13421-016-0614-y
Driver, R. (1981) Pupils’ alternative framework in science. European Journal of Science Education, 3(1), 93-101. https://doi.org/10.1080/0140528810030109
Ernest, P. (1991). The Philosophy of Mathematics Education. London: Routledge.
Ernest, P. (2011). The Psychology of Learning Mathematics: The Cognitive, Affective and Contextual Domains of Mathematics Education. Saarbrücken, LAP Lambert Academic Publ.
Francisco, J. M. (2013). Learning in collaborative settings: Students building on each other’s ideas to promote their mathematical understanding. Educational Studies in Mathematics, 82(3), 417-438. https://doi.org/10.1007/s10649-012-9437-3
Gray, E. M., & Tall, D. O. (1994). Duality, ambiguity, and flexibility: A “proceptual” view of simple arithmetic. Journal for research in Mathematics Education, 25(2), 116-140. https://doi.org/10.2307/749505
Heinze, A., Star, J. R., & Verschaffel, L. (2009). Flexible and adaptive use of strategies and representations in mathematics education. ZDM Mathematics Education, 41, 535-540. https://doi.org/10.1007/s11858-0090214-4
Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for research in mathematics education, 28(5), 524-549. https://doi.org/10.2307/749690
Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 65-97). Macmillan Publishing Co, Inc.
Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1-27). Hillsdale, NJ: Lawrence Erlbaum Associates.
Jonassen, D. H., Reeves, T. C., Hong, N., Harvey, D., & Peters, K. (1997). Concept mapping as cognitive learning and assessment tools. Journal of interactive learning research, 8(3), 289.
Kadijevich, D., & Haapasalo, L. (2001). Linking procedural and conceptual mathematical knowledge through CAL. Journal of Computer Assisted Learning, 17(2), 156-165. https://doi.org/10.1046/j.0266-4909.2001.00168.x
Kazemi, E., & Stipek, D. (2001). Promoting conceptual thinking in four upper-elementary mathematics classrooms. Elementary School Journal, 102, 59-80. https://doi.org/10.1086/499693
Kelly, G. A. (1955). The Psychology of Personal Constructs: Volume One: Theory and Personality. London: Routledge.
Kinchin, I. M., Hay, D. B., & Adams, A. (2000). How a qualitative approach to concept map analysis can be used to aid learning by illustrating patterns of conceptual development. Educational research, 42(1), 43-57. https://doi.org/10.1080/001318800363908
Lambiotte, J. G., & Dansereau, D. F. (1991). Effects of knowledge maps and prior knowledge on recall of science lecture content. The Journal of Experimental Education, 60(3), 189-201. https://doi.org/10.1080/00220973.1992.9943875
Lanier, P. (1997). Assesment in the Service of Instruction [Paper presentation]. Annual meeting of the National of Supervisors of Mathematics, St.Paul, MN.
Laturno, J. (1993). The validity of concept maps as a research tool in remedial college mathematics. Proceedings of the 16th Annual Meeting of the Group for the Psychology of Mathematics Education, North American Chapter (pp. 60-66). Baton Rouge, LA.
Lewis, A., & Smith, D. (1993). Defining higher order thinking. Theory into practice, 32(3), 131-137. https://doi.org/10.1080/00405849309543588
Liu, X. (2009). Essentials of science classroom assessment. Sage Publications. https://doi.org/10.4135/9781483349442
McGowen, M. A., & Davis, G. E. (2019). Spectral analysis of concept maps of high and low gain undergraduate mathematics students. The Journal of Mathematical Behavior, 55, 100686. https://doi.org/10.1016/j.jmathb.2019.01.002
McGowen, M., & Tall, D. (1999). Concept Maps & Schematic Diagrams as Devices for Documenting the Growth of Mathematical Knowledge. The 23rd International Group for the Psychology of Mathematics Education (Vol. 3). Haifa, Israel.
Michener, E. R. (1978). Understanding understanding mathematics. Cognitive science, 2(4), 361-383. https://doi.org/10.1207/s15516709cog0204_3
Miller, S. P., & Hudson, P. J. (2007). Using evidence‐based practices to build mathematics competence related to conceptual, procedural, and declarative knowledge. Learning Disabilities Research & Practice, 22(1), 47-57. https://doi.org/10.1111/j.1540-5826.2007.00230.x
Moreira, M. A. (1979). Concept Maps as Tools for Teaching. Journal of College Science Teaching, 8(5), 283-86.
Novak, J. D. (1990). Concept mapping: A useful tool for science education. Journal of research in science teaching, 27(10), 937-949. https://doi.org/10.1002/tea.3660271003
Park, K. & Travers, K. (1996). A Comparative Study of a Computer-Based and a Standard College First-Year Calculus Course. In E.Dubinsky, A. Schoenfeld, & J. Kaput (Eds.), Research in Collegiate Mathematics Education (pp. 155-176). Providence, RI: American Mathematical Society. https://doi.org/10.1090/cbmath/006/07
Peled, I., & Segalis, B. (2005). It’s not too late to conceptualize: Constructing a generalized subtraction schema by abstracting and connecting procedures. Mathematical Thinking and Learning, 7(3), 207-230. https://doi.org/10.1207/s15327833mtl0703_2
Pfund, H., & Duit, R. (1994) Bibliography - Students’ Alternative Frameworks and Science Education (4th edn). Kiel: Institute for Science Education.
Piaget, J., & Cook, M. (1952). The origins of intelligence in children (Vol. 8, No. 5, p. 18). New York: International Universities Press. https://doi.org/10.1037/11494-000
Reinfried, S. (2006). Conceptual change in Physical Geography and Environmental Sciences through mental model building: The example of groundwater. International Research in Geographical and Environmental Education, 15(1), 41-61. https://doi.org/10.2167/irgee186.0
Rittle-Johnson, B., & Alibali, M. W. (1999). Conceptual and procedural knowledge of mathematics: Does one lead to the other? Journal of Educational Psychology, 91(1), 175-189. https://doi.org/10.1037/0022-0663.91.1.175
Rumelhart, D. E., & Norman, D. A. (1978). Accretion, tuning, and restructuring: Three modes of learning. In J. W. Cotton & R. Klatzky (Eds.), Semantic factors in cognition (pp. 37-53). Hillsdale, NJ: Lawrence Erlbaum.
Schneider, M., Rittle-Johnson, B., & Star, J. R. (2011). Relations among conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge. Developmental Psychology, 47, 1525-1538. https://doi.org/10.1037/a0024997
Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational studies in mathematics, 22(1), 1-36. https://doi.org/10.1007/BF00302715
Siegler, R. S., & Crowley, K. (1994). Constraints on learning in nonprivileged domains. Cognitive Psychology, 27, 194-226. https://doi.org/10.1006/cogp.1994.1016
Skemp, R. (1976). Instrumental understanding and relational understanding. Mathematics Teaching, 77, 20-26.
Skemp, R. R. (1979). Intelligence, Learning, and Action. a New Model for Theory and Practice in Education: New York Wiley.
Skemp, R.R. (1986). The Psychology of Learning Mathematics. London: Penguin Books
Smith, R. O. (2014). Beyond passive learning: Problem-based learning and concept maps to promote basic and higher-order thinking in basic skills instruction. Journal of Research and Practice for Adult Literacy, Secondary, and Basic Education, 3(2), 50-55.
Soliman, M., & Hilal, A. (2016). Investigating the effects of Computer-Assisted Instruction on Achievement and Attitudes towards Mathematics Among Seventh-Grade Students in Kuwait. International Journal of Technology in Mathematics Education, 23(4), 145-160.
Swan, M. (2005). Standards Unit-Improving learning in mathematics: challenges and strategies. DfES. Retrieved from http://www.Ncetm.org.uk/files/224/improving_learning_in_mathematicsi.pdf
Tall, D. (1994b). Understanding the processes of advanced mathematical thinking [Invited ICMI lecture]. International Congress of Mathematicians, Zurich.
Thomas, G. (2011). A typology for the case study in social science following a review of definition, discourse, and structure. Qualitative inquiry, 17(6), 511-521. https://doi.org/10.1177/1077800411409884
Verschaffel, L., Luwel, K., Torbeyns, J., & Van Dooren, W. (2009). Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education. European Journal of Psychology of Education, 24, 335-359. https://doi.org/10.1007/BF03174765
Von Glasersfeld, E. (1998). Cognition, construction of knowledge, and teaching. In Constructivism in science education (pp. 11-30). Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5032-3_2
Vygotsky, L. S. (1978). Mind in Society. Cambridge: Harvard University Press. In E. Wenger, N. White, & J. Smith (2009), Digital habitats: Stewarding technology for communities. CP Square Press.
Watt, D. (2007). On becoming a qualitative researcher: the value of reflexivity. Qualitative Report, 12(1), 82-101.
Wilkerson-Jerde, M. H., & Wilensky, U. J. (2011). How do mathematicians learn math?: Resources and acts for constructing and understanding mathematics. Educational Studies in Mathematics, 78(1), 21. https://doi.org/10.1007/s10649-011-9306-5
Williams, C. G. (1998). Using concept maps to assess conceptual knowledge of function. Journal for Research in Mathematics Education, 414-421. https://doi.org/10.2307/749858

How to cite this article

APA

Hammad, S., Dimitriadis, C., & Graham, T. (2021). Using Concepts Maps in a Foundation Mathematics Course: What Have we Learnt?. Eurasia Journal of Mathematics, Science and Technology Education, 17(2), em1943. https://doi.org/10.29333/ejmste/9700

Vancouver

Hammad S, Dimitriadis C, Graham T. Using Concepts Maps in a Foundation Mathematics Course: What Have we Learnt?. EURASIA J Math Sci Tech Ed. 2021;17(2):em1943. https://doi.org/10.29333/ejmste/9700

AMA

Hammad S, Dimitriadis C, Graham T. Using Concepts Maps in a Foundation Mathematics Course: What Have we Learnt?. EURASIA J Math Sci Tech Ed. 2021;17(2), em1943. https://doi.org/10.29333/ejmste/9700

Chicago

Hammad, Sali, Christos Dimitriadis, and Ted Graham. "Using Concepts Maps in a Foundation Mathematics Course: What Have we Learnt?". Eurasia Journal of Mathematics, Science and Technology Education 2021 17 no. 2 (2021): em1943. https://doi.org/10.29333/ejmste/9700

Harvard

Hammad, S., Dimitriadis, C., and Graham, T. (2021). Using Concepts Maps in a Foundation Mathematics Course: What Have we Learnt?. Eurasia Journal of Mathematics, Science and Technology Education, 17(2), em1943. https://doi.org/10.29333/ejmste/9700

MLA

Hammad, Sali et al. "Using Concepts Maps in a Foundation Mathematics Course: What Have we Learnt?". Eurasia Journal of Mathematics, Science and Technology Education, vol. 17, no. 2, 2021, em1943. https://doi.org/10.29333/ejmste/9700